Chapter 11: Analyzing the Association Between Categorical Variables

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Chapter 11 Analyzing the Association Between
Categorical Variables

• Section 11.1 What is Independence and What is
  Association?

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Learning Objectives

1. Comparing Percentages
2. Independence vs. Dependence

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Learning Objective 1 Example Is There an
Association Between Happiness and Family Income?
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Learning Objective 1 Example Is There an
Association Between Happiness and Family Income?

• The percentages in a particular row of a table
  are called conditional percentages
• They form the conditional distribution for
  happiness, given a particular income level

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Learning Objective 1 Example Is There an
Association Between Happiness and Family Income?
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Learning Objective 1 Example Is There an
Association Between Happiness and Family Income?

• Guidelines when constructing tables with
  conditional distributions
• Make the response variable the column variable
• Compute conditional proportions for the response
  variable within each row
• Include the total sample sizes

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Learning Objective 2Independence vs. Dependence

• For two variables to be independent, the
  population percentage in any category of one
  variable is the same for all categories of the
  other variable
• For two variables to be dependent (or
  associated), the population percentages in the
  categories are not all the same

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Learning Objective 2Independence vs. Dependence

• Are race and belief in life after death
  independent or dependent?
• The conditional distributions in the table are
  similar but not exactly identical
• It is tempting to conclude that the variables are
  dependent

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Learning Objective 2Independence vs. Dependence

• Are race and belief in life after death
  independent or dependent?
• The definition of independence between variables
  refers to a population
• The table is a sample, not a population

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Learning Objective 2Independence vs. Dependence

• Even if variables are independent, we would not
  expect the sample conditional distributions to be
  identical
• Because of sampling variability, each sample
  percentage typically differs somewhat from the
  true population percentage

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Chapter 11 Analyzing the Association Between
Categorical Variables

• Section 11.2 How Can We Test Whether Categorical
  Variables Are Independent?

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Learning Objectives

1. A Significance Test for Categorical Variables
2. What Do We Expect for Cell Counts if the
   Variables Are Independent?
3. How Do We Find the Expected Cell Counts?
4. The Chi-Squared Test Statistic
5. The Chi-Squared Distribution
6. The Five Steps of the Chi-Squared Test of
   Independence

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Learning Objectives

1. Chi-Squared is Also Used as a Test of
   Homogeneity
2. Chi-Squared and the Test Comparing Proportions in
   2x2 Tables
3. Limitations of the Chi-Squared Test

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Learning Objective 1A Significance Test for
Categorical Variables

• Create a table of frequencies divided into the
  categories of the two variables
• The hypotheses for the test are
• H0 The two variables are independent
• Ha The two variables are dependent
  (associated)
• The test assumes random sampling and a large
  sample size (cell counts in the frequency table
  of at least 5)

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Learning Objective 2What Do We Expect for Cell
Counts if the Variables Are Independent?

• The count in any particular cell is a random
  variable
• Different samples have different count values
• The mean of its distribution is called an
  expected cell count
• This is found under the presumption that H0 is
  true

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Learning Objective 3How Do We Find the Expected
Cell Counts?

• Expected Cell Count
• For a particular cell,
• The expected frequencies are values that have the
  same row and column totals as the observed
  counts, but for which the conditional
  distributions are identical (this is the
  assumption of the null hypothesis).

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Learning Objective 3How Do We Find the Expected
Cell Counts?Example
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Learning Objective 4The Chi-Squared Test
Statistic

• The chi-squared statistic summarizes how far the
  observed cell counts in a contingency table fall
  from the expected cell counts for a null
  hypothesis

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Learning Objective 4Example Happiness and
Family Income

• State the null and alternative hypotheses for
  this test
• H0 Happiness and family income are independent
• Ha Happiness and family income are dependent
  (associated)

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Learning Objective 4Example Happiness and
Family Income

• Report the statistic and explain how it was
  calculated
• To calculate the statistic, for each cell,
  calculate
• Sum the values for all the cells
• The value is 73.4

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Learning Objective 4Example Happiness and
Family Income
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Learning Objective 4The Chi-Squared Test
Statistic

• The larger the value, the greater the
  evidence against the null hypothesis of
  independence and in support of the alternative
  hypothesis that happiness and income are
  associated

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Learning Objective 5The Chi-Squared Distribution

• To convert the test statistic to a
  P-value, we use the sampling distribution of the
  statistic
• For large sample sizes, this sampling
  distribution is well approximated by the
  chi-squared probability distribution

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Learning Objective 5The Chi-Squared Distribution
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Learning Objective 5The Chi-Squared Distribution

• Main properties of the chi-squared distribution
• It falls on the positive part of the real number
  line
• The precise shape of the distribution depends on
  the degrees of freedom
• df (r-1)(c-1)

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Learning Objective 5The Chi-Squared Distribution

• Main properties of the chi-squared distribution
• The mean of the distribution equals the df value
• It is skewed to the right
• The larger the value, the greater the
  evidence against H0 independence

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Learning Objective 5The Chi-Squared Distribution
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Learning Objective 6The Five Steps of the
Chi-Squared Test of Independence

• 1. Assumptions
• Two categorical variables
• Randomization
• Expected counts 5 in all cells

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Learning Objective 6The Five Steps of the
Chi-Squared Test of Independence

• 2. Hypotheses
• H0 The two variables are independent
• Ha The two variables are dependent (associated)

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Learning Objective 6The Five Steps of the
Chi-Squared Test of Independence

• 3. Test Statistic

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Learning Objective 6The Five Steps of the
Chi-Squared Test of Independence

• 4. P-value Right-tail probability above the
  observed value, for the chi-squared
  distribution with df (r-1)(c-1)
• 5. Conclusion Report P-value and interpret in
  context
• If a decision is needed, reject H0 when P-value
  significance level

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Learning Objective 7Chi-Squared is Also Used as
a Test of Homogeneity

• The chi-squared test does not depend on which is
  the response variable and which is the
  explanatory variable
• When a response variable is identified and the
  population conditional distributions are
  identical, they are said to be homogeneous
• The test is then referred to as a test of
  homogeneity

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Learning Objective 8Chi-Squared and the Test
Comparing Proportions in 2x2 Tables

• In practice, contingency tables of size 2x2 are
  very common. They often occur in summarizing the
  responses of two groups on a binary response
  variable.
• Denote the population proportion of success by p1
  in group 1 and p2 in group 2
• If the response variable is independent of the
  group, p1p2, so the conditional distributions
  are equal
• H0 p1p2 is equivalent to H0 independence

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Learning Objective 8Example Aspirin and Heart
Attacks Revisited
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Learning Objective 8 Example Aspirin and
Heart Attacks Revisited

• What are the hypotheses for the chi-squared test
  for these data?
• The null hypothesis is that whether a doctor has
  a heart attack is independent of whether he takes
  placebo or aspirin
• The alternative hypothesis is that theres an
  association

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Learning Objective 8 Example Aspirin and
Heart Attacks Revisited

• Report the test statistic and P-value for the
  chi-squared test
• The test statistic is 25.01 with a P-value of
  0.000
• This is very strong evidence that the population
  proportion of heart attacks differed for those
  taking aspirin and for those taking placebo

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Learning Objective 8 Example Aspirin and
Heart Attacks Revisited

• The sample proportions indicate that the aspirin
  group had a lower rate of heart attacks than the
  placebo group

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Learning Objective 9Limitations of the
Chi-Squared Test

• If the P-value is very small, strong evidence
  exists against the null hypothesis of
  independence
• But
• The chi-squared statistic and the P-value tell us
  nothing about the nature of the strength of the
  association

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Learning Objective 9Limitations of the
Chi-Squared Test

• We know that there is statistical significance,
  but the test alone does not indicate whether
  there is practical significance as well

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Learning Objective 9Limitations of the
Chi-Squared Test

• The chi-squared test is often misused. Some
  examples are
• when some of the expected frequencies are too
  small
• when separate rows or columns are dependent
  samples
• data are not random
• quantitative data are classified into categories
  - results in loss of information

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Learning Objective 10Goodness of Fit
Chi-Squared Tests

• The Chi-Squared test can also be used for testing
  particular proportion values for a categorical
  variable.
• The null hypothesis is that the distribution of
  the variable follows a given probability
  distribution the alternative is that it does not
• The test statistic is calculated in the same
  manner where the expected counts are what would
  be expected in a random sample from the
  hypothesized probability distribution
• For this particular case, the test statistic is
  referred to as a goodness-of-fit statistic.

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Chapter 11 Analyzing the Association Between
Categorical Variables

• Section 11.3 How Strong is the Association?

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Learning Objectives

1. Analyzing Contingency Tables
2. Measures of Association
3. Difference of Proportions
4. The Ratio of Proportions Relative Risk
5. Properties of the Relative Risk
6. Large Chi-square Does Not Mean Theres a Strong
   Association

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Learning Objective 1Analyzing Contingency Tables

• Is there an association?
• The chi-squared test of independence addresses
  this
• When the P-value is small, we infer that the
  variables are associated

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Learning Objective 1Analyzing Contingency Tables

• How do the cell counts differ from what
  independence predicts?
• To answer this question, we compare each observed
  cell count to the corresponding expected cell
  count

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Learning Objective 1Analyzing Contingency Tables

• How strong is the association?
• Analyzing the strength of the association reveals
  whether the association is an important one, or
  if it is statistically significant but weak and
  unimportant in practical terms

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Learning Objective 2Measures of Association

• A measure of association is a statistic or a
  parameter that summarizes the strength of the
  dependence between two variables
• a measure of association is useful for comparing
  associations

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Learning Objective 3Difference of Proportions

• An easily interpretable measure of association is
  the difference between the proportions making a
  particular response

Case (a) exhibits the weakest possible
association no association. The difference of
proportions is 0
Case (b) exhibits the strongest possible
association The difference of proportions is 1
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Learning Objective 3Difference of Proportions

• In practice, we dont expect data to follow
  either extreme (0 difference or 100
  difference), but the stronger the association,
  the larger the absolute value of the difference
  of proportions

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Learning Objective 3Difference of Proportions
Example Do Student Stress and Depression Depend
on Gender?

• Which response variable, stress or depression,
  has the stronger sample association with gender?
• The difference of proportions between females and
  males was 0.35 0.16 0.19 for feeling stressed
• The difference of proportions between females and
  males was 0.08 0.06 0.02 for feeling depressed

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Learning Objective 3Difference of Proportions
Example Do Student Stress and Depression Depend
on Gender?

• In the sample, stress (with a difference of
  proportions 0.19) has a stronger association
  with gender than depression has (with a
  difference of proportions 0.02)

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Learning Objective 4The Ratio of Proportions
Relative Risk

• Another measure of association, is the ratio of
  two proportions p1/p2
• In medical applications in which the proportion
  refers to an adverse outcome, it is called the
  relative risk

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Learning Objective 4 Example Relative Risk
for Seat Belt Use and Outcome of Auto Accidents

• Treating the auto accident outcome as the
  response variable, find and interpret the
  relative risk

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Learning Objective 4 Example Relative Risk
for Seat Belt Use and Outcome of Auto Accidents

• The adverse outcome is death
• The relative risk is formed for that outcome
• For those who wore a seat belt, the proportion
  who died equaled 510/412,878 0.00124
• For those who did not wear a seat belt, the
  proportion who died equaled 1601/164,128
  0.00975

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Learning Objective 4 Example Relative Risk
for Seat Belt Use and Outcome of Auto Accidents

• The relative risk is the ratio
• 0.00124/0.00975 0.127
• The proportion of subjects wearing a seat belt
  who died was 0.127 times the proportion of
  subjects not wearing a seat belt who died

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Learning Objective 4 Example Relative Risk
for Seat Belt Use and Outcome of Auto Accidents

• Many find it easier to interpret the relative
  risk but reordering the rows of data so that the
  relative risk has value above 1.0

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Learning Objective 4 Example Relative Risk
for Seat Belt Use and Outcome of Auto Accidents

• Reversing the order of the rows, we calculate the
  ratio
• 0.00975/0.00124 7.9
• The proportion of subjects not wearing a seat
  belt who died was 7.9 times the proportion of
  subjects wearing a seat belt who died

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Learning Objective 4 Example Relative Risk
for Seat Belt Use and Outcome of Auto Accidents

• A relative risk of 7.9 represents a strong
  association
• This is far from the value of 1.0 that would
  occur if the proportion of deaths were the same
  for each group
• Wearing a set belt has a practically significant
  effect in enhancing the chance of surviving an
  auto accident

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Learning Objective 5Properties of the Relative
Risk

• The relative risk can equal any nonnegative
  number
• When p1 p2, the variables are independent and
  relative risk 1.0
• Values farther from 1.0 (in either direction)
  represent stronger associations

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Learning Objective 6Large Does Not Mean
Theres a Strong Association

• A large chi-squared value provides strong
  evidence that the variables are associated
• It does not imply that the variables have a
  strong association
• This statistic merely indicates (through its
  P-value) how certain we can be that the variables
  are associated, not how strong that association is

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Chapter 11 Analyzing the Association Between
Categorical Variables

• Section 11.4 How Can Residuals Reveal The
  Pattern of Association?

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Learning Objectives

1. Association Between Categorical Variables
2. Residual Analysis

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Learning Objective 1Association Between
Categorical Variables

• The chi-squared test and measures of association
  such as (p1 p2) and p1/p2 are fundamental
  methods for analyzing contingency tables
• The P-value for summarized the strength of
  evidence against H0 independence

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Learning Objective 1Association Between
Categorical Variables

• If the P-value is small, then we conclude that
  somewhere in the contingency table the population
  cell proportions differ from independence
• The chi-squared test does not indicate whether
  all cells deviate greatly from independence or
  perhaps only some of them do so

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Learning Objective 2Residual Analysis

• A cell-by-cell comparison of the observed counts
  with the counts that are expected when H0 is true
  reveals the nature of the evidence against H0
• The difference between an observed and expected
  count in a particular cell is called a residual

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Learning Objective 2Residual Analysis

• The residual is negative when fewer subjects are
  in the cell than expected under H0
• The residual is positive when more subjects are
  in the cell than expected under H0

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Learning Objective 2Residual Analysis

• To determine whether a residual is large enough
  to indicate strong evidence of a deviation from
  independence in that cell we use a adjusted form
  of the residual the standardized residual

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Learning Objective 2Residual Analysis

• The standardized residual for a cell
• (observed count expected count)/se
• A standardized residual reports the number of
  standard errors that an observed count falls from
  its expected count
• The se describes how much the difference would
  tend to vary in repeated sampling if the
  variables were independent
• Its formula is complex
• Software can be used to find its value
• A large standardized residual value provides
  evidence against independence in that cell

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Learning Objective 2 Example Standardized
Residuals for Religiosity and Gender

• To what extent do you consider yourself a
  religious person?

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Learning Objective 2 Example Standardized
Residuals for Religiosity and Gender

• Interpret the standardized residuals in the table
  
• The table exhibits large positive residuals for
  the cells for females who are very religious and
  for males who are not at all religious.
• In these cells, the observed count is much larger
  than the expected count
• There is strong evidence that the population has
  more subjects in these cells than if the
  variables were independent

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Learning Objective 2 Example Standardized
Residuals for Religiosity and Gender

• The table exhibits large negative residuals for
  the cells for females who are not at all
  religious and for males who are very religious
• In these cells, the observed count is much
  smaller than the expected count
• There is strong evidence that the population has
  fewer subjects in these cells than if the
  variables were independent

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Chapter 11 Analyzing the Association Between
Categorical Variables

• Section 11.5 What if the Sample Size is Small?
• Fishers Exact Test

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Learning Objectives

1. Fishers Exact Test
2. Example using Fishers Exact Test
3. Summary of Fishers Exact Test of Independence
   for 2x2 Tables

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Learning Objective 1Fishers Exact Test

• The chi-squared test of independence is a
  large-sample test
• When the expected frequencies are small, any of
  them being less than about 5, small-sample tests
  are more appropriate
• Fishers exact test is a small-sample test of
  independence

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Learning Objective 1Fishers Exact Test

• The calculations for Fishers exact test are
  complex
• Statistical software can be used to obtain the
  P-value for the test that the two variables are
  independent
• The smaller the P-value, the stronger the
  evidence that the variables are associated

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Learning Objective 2Fishers Exact Test
Example Tea Tastes Better with Milk Poured First?

• This is an experiment conducted by Sir Ronald
  Fisher
• His colleague, Dr. Muriel Bristol, claimed that
  when drinking tea she could tell whether the milk
  or the tea had been added to the cup first

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Learning Objective 2Fishers Exact Test
Example Tea Tastes Better with Milk Poured First?

• Experiment
• Fisher asked her to taste eight cups of tea
• Four had the milk added first
• Four had the tea added first
• She was asked to indicate which four had the milk
  added first
• The order of presenting the cups was randomized

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Learning Objective 2Fishers Exact Test
Example Tea Tastes Better with Milk Poured First?

• Results

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Learning Objective 2Fishers Exact Test
Example Tea Tastes Better with Milk Poured First?

• Analysis

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Learning Objective 2Fishers Exact Test
Example Tea Tastes Better with Milk Poured First?

• The one-sided version of the test pertains to the
  alternative that her predictions are better than
  random guessing
• Does the P-value suggest that she had the ability
  to predict better than random guessing?

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Learning Objective 2Fishers Exact Test
Example Tea Tastes Better with Milk Poured First?

• The P-value of 0.243 does not give much evidence
  against the null hypothesis
• The data did not support Dr. Bristols claim that
  she could tell whether the milk or the tea had
  been added to the cup first

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Learning Objective 3Summary of Fishers Exact
Test of Independence for 2x2 Tables

• Assumptions
• Two binary categorical variables
• Data are random
• Hypotheses
• H0 the two variables are independent (p1p2)
• Ha the two variables are associated
• (p1?p2 or p1gtp2 or p1ltp2)

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Learning Objective 3Summary of Fishers Exact
Test of Independence for 2x2 Tables

• Test Statistic
• First cell count (this determines the others
  given the margin totals)
• P-value
• Probability that the first cell count equals the
  observed value or a value even more extreme as
  predicted by Ha
• Conclusion
• Report the P-value and interpret in context. If
  a decision is required, reject H0 when P-value
  significance level